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First Step:
The first step is to write down all the basic partial differential
equations of classical physics which relate to the problem of interest.
Do not write down equations containing second space derivatives
which are derived from firstderivative equations.
Write down the firstderivative equations.
Write each component of vector or matrix equations.
In acoustics we have the gradient pressure p
gives rise to an acceleration of mass density .For convenience we restrict motion to the x, z plane.
Letting u and w represent x and z components of velocity we have
 
(1) 
 
 (2) 
Another equation which is important in acoustics
is the one that states that the divergence of velocity
multiplied by the incompressibility K yields the rate of pressure decrease.
 
(3) 
In (913) we included a pressure source s.
This is something to be externally prescribed.
The quantity s may be a source of chemical energy such as an explosion;
thus it may vanish everywhere except at a point.
Distributed sources are also often of interest;
for example,
radioactive rocks in a heatflow model of the earth.
To be more general,
we could also have put momentum sources into (911) or (912),
but the basic principles will be adequately exemplified with a source
only in (913).
Second step:
The wave disturbance variables are taken to be unknown
and the material properties known.
Count equations and unknowns.
We have three equations;
u, w, and p are the three unknowns.
We take K, , and s to be known.
Notice that the equations are linear in the unknowns.
Now we make the stratification assumption;
that is,
we assume K and are functions of depth z
only and that they are constant in x.
Since our linear equations now have constant coefficients
with respect to x and t,
we may always expect sinusoidal solutions in x and t.
We do not know what to expect of our solutions
in the z coordinate because of the arbitrary zdependence
of the coefficients K and .This leads to step three.
Third step:
Fourier transform time and the space coordinates with constant coefficients.
In other words, we make the following substitution into (911),
(912), and (913)
 
(4) 
After substitution, cancel the exponential and obtain
 
(1) 
 (2) 
 (3) 
Fourth step:
Eliminate algebraically the algebraic unknowns.
In other words, when you examine (915) you see terms
in but you do not see .This means that U is an algebraic variable which can be eliminated
by purely algebraic means.
We do this by substituting (915a), into (915c).
Fifth step:
Bring terms to the left,
bring all others to the right,
and arrange terms into a neat matrix form. We have
and then
 
(6) 
Sixth step:
Recognize that,
no matter the physical problem with which you started,
you should have a matrix firstorder differential equation of the form
 
(7) 
where is a vector containing the field variables of interest,
is a matrix depending on temporal and spatial frequency
and on material properties,
and is a (possibly absent) vector function of the sources.
Before we look into techniques of solving (917)
we can immediately deduce that in a sourcefree region
the field variables are smoother functions
than the material properties.
To see this, consider two homogeneous layers in contact.
At the contact the matrix has stepfunction discontinuities.
Now let us see whether the wave fields in can have stepfunction discontinuities.
Obviously they cannot,
since a step discontinuity
in would imply ,whereas (917) in a sourcefree region states
that and both
and are supposed finite.
This does not mean that all field variables are always smooth.
The algebraic variables eliminated in the fourth step
can and often will be discontinuous at layer boundaries.
EXERCISES:

What form does (917) take for the heatflow equation?
Include radioactive sources.
[HINT: See equations (1011) and (1012).]

Using Maxwell's equations,
,,and Ohm's law,
where is conductivity,
set and derive (917).

In electrostatics the electric field in the ionosphere
may be derived from a potential , the
divergence of electrical current vanishes and Ohm's law must have an extra term due to wind
(a current source due to differential drag on ions and electrons
across the earth's magnetic field)
.Assume you know .What form does (917) take assuming and to be scalars?
Indicate how the calculation proceeds if and are matrices (assume you have the inverse of any matrix you wish).

In magnetostatics
and , and .Taking as given, what is the form of (917)?

This exercise illustrates the linearization of nonlinear problems.
For acoustic waves in a stratified windy atmosphere we used the trial solutions
Reduce the partial differential equations
to a matrix ordinary differential equation.
HINT: The horizontal acceleration terms is
with a like term for vertical acceleration.
Drop secondorder terms in , , and .

Two equations come from heat flow:
(H_{x}, H_{z}) equals the conductivity multiplied by
the negative of the temperature gradient
.The time derivative of temperature multiplied by the heat capacity c
equals the negative of the heatflow divergence
gives another equation.
Insert the trial solutions
 (a)
 First derive steadystate equations for and assuming and vanish.
 (b)
 Assuming and satisfy part (a), find equations for and .
 (c)
 Repeat (a) and (b) assuming linear temperature dependence of heat capacity and conductivity, i.e.,
You will have to drop squared terms in and .

Consider a compressible liquid sphere pulsating radially
under its own gravitational attraction.
What is the form of (916)?
Next: NUMERICAL MATRIZANTS
Up: Mathematical physics in stratified
Previous: Mathematical physics in stratified
Stanford Exploration Project
10/30/1997